Random free-product extension conjecture for stable subgroups of right-angled Artin groups

Let $A(\Gamma)$ be a right-angled Artin group whose defining graph $\Gamma$ is connected and not a join, and let $H$ be a stable subgroup of $A(\Gamma)$. Let $(\mu_i)$ be a sequence of permissible probability distributions on $A(\Gamma)$, and let $R$ be the associated random subgroup. Stable-subgroup extension conjecture. The subgroup generated by $H$ and $R$ should satisfy

$$\langle H,R\rangle\cong H\ast R$$

and should be stable in $A(\Gamma)$. This conjecture predicts that the random free-product construction established for convex cocompact subgroups in mapping class groups extends to stable subgroups of right-angled Artin groups; its status is open.